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Cascade of period doubling bifurcations and large stochasticity in the motions of a compass
V. Croquette, Christine Poitou
To cite this version:
V. Croquette, Christine Poitou. Cascade of period doubling bifurcations and large stochasticity in the motions of a compass. Journal de Physique Lettres, Edp sciences, 1981, 42 (24), pp.537-539.
�10.1051/jphyslet:019810042024053700�. �jpa-00231994�
L-537
Cascade of period doubling bifurcations and large stochasticity
in the motions of a compass (*)
V. Croquette and C. Poitou
Service de Physique du Solide et de Résonance Magnétique, CEN Saclay, 91191 Gif sur Yvette Cedex, France (Re~u le 3 juin 1981, accepte le 27 octobre 1981)
Résumé. 2014 Nous proposons un système mécanique très simple, constitué d’une boussole dans un champ alternatif périodique, qui permet l’observation de la stochasticité à grande échelle des systèmes hamiltoniens. Nous montrons que ce système présente aussi une cascade de dédoublements de période. Nous déterminons expérimentalement
les seuils des premiers dédoublements.
Abstract. 2014 We propose a very simple mechanical device, consisting of a compass placed in a periodic oscillating field, which renders the observation of large scale stochasticity in Hamiltonian systems possible. We show that this device also exhibits a cascade of period doubling bifurcations. We determine experimentally the threshold of the first doublings.
J. Physique - LETTRES 42 (1981) L-537 - L-539 15 DÉCEMBRE 1981,
Classification Physics Abstracts 46.90
If we place a compass, initially neglecting friction,
in an oscillating magnetic field, perpendicular to the
compass axis, we have a simple mechanical device,
which models a well-known system : the synchronous bipolar motor. We might think that the possible
motions of this compass are naturally a clockwise or
a counterclockwise rotation. In fact, by doing so, we
have not considered the non-linear property of this device, and if these two kinds of motion really exist,
a multitude of different motions are also possible,
some periodic, others chaotic.
For this device, the angle of rotation 0 is governed by the equation :
with F = (M. Bo/2 J ) and where M and J are res- pectively the magnetic and inertial momentum of the
compass. Bo is the magnitude of the magnetic field, oscillating at the frequency (co/2 n). A detailed study
of this kind of equation (1) has been performed by
D. Escande and F. Doveil [1] and also in [6, 7].
Let us first consider the compass placed in a single rotating field. In the rotating frame, equation (1)
becomes :
81 = - F [sin 61 ] (2)
(*) La version francaisc de cet article a été proposée pour publi-
cation aux Comptes Rendus de 1’Academie des Sciences.
e) Equation (1) also models a phase lock loop in electronics.
which is exactly the completely-integrable equation of
the pendulum [2]. It has two kinds of solutions : oscillations in which the compass is locked with and oscillates around the rotating field with its own fre- quency coo ( = F when the magnitude of these oscil- lations is small) and rotations in which the compass is rotating at a rate different from that of the field.
In phase space (0, 0), the lock-in or the resonance is associated with a set of closed trajectories (ellipsoidal
for small oscillations), the largest closed trajectory being the separatrix. Similar results will be obtained with a second rotating field whose resonance will be
symmetric with the first one about the axis 0=0.
If we now consider the two fields simultaneously, we easily understand that two fundamental resonances
will appear in phase space, symmetric about the axis
0=0. However, this is only true as long as the per- turbation level is small, the system is no longer inte- grable, the phase space becomes three dimensional and stochastic trajectories appear. It is possible to
return to the former phase space (0, 0), through a
Poincare transformation. The K.A.M. theorem [3]
means that the two resonances that we have already
described remain stable as long as the perturbation
level is not too high. The perturbation level is defined by the stochasticity parameter s = (2~/F/(D) which
measures the width, in phase space, of the resonances
compared with their separation. When s = 1, the separatrix of the two unperturbed resonances touch each other, the resonances overlap, this is the criterion
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyslet:019810042024053700
L-538 JOURNAL DE PHYSIQUE - LETTRES
,proposed by Chirikov [2], for the occurrence of
large scale stochasticity.
The trajectories in the phases space at s = 0.68 are given in figure 1; we find a number of resonances of which the larger A and A’ are just those we have
described. The resonances B and B: correspond to
oscillations of the compass with ~0~=0. On the
other hand, the phase space also contains diffuse areas,
growing like s at the expense of the resonances, which also combine so allowing the system to diffuse into
a larger and larger areas of phase space. When the system passes from resonance A to resonance A’, large
scale stochasticity occurs.
Fig. 1. - Calculated shape of the phases space at s = 0.68. (We thank D. Escande and F. Doveil for this document.)
1. Experimental lay-out. - A compass, fixed on an
axis guided by two bearings, is placed between two
coils in a Helmholtz configuration fed by an alter- nating current. The stochasticity parameter s is fixed
by adjusting the magnitude of this current. The data acquisition system consists of a pick-up coil placed
near the compass perpendicular to the excitation coils.
The signal so induced is then Fourier analysed.
An important feature of our device is that it is a dissipative system. However, the friction is small, and the contraction, in phase space, slow so that we will be able to draw conclusions on the phase space of the associated Hamiltonian system, provided that the
« trajectories » of the compass are considered over a
short time interval. It is possible to model (2) our device
with the equation :
with (oc/6o) ~ 3 x 10-2.
2. Study of the destabilization of resonances. - In this study, the dissipative property plays a fundamen-
tal role : it allows an automatic centring on the ellip-
tical point associated with each resonance to be
achieved, so that it will be possible to follow the
bifurcation cascade associated with the destabilization of these resonances.
Let us consider, to begin with the fundamental
resonances A and A’. They correspond to a synchro-
nous rotation with one of the rotating fields. The second rotating field acts as a perturbation at the frequency (2 w/2 n). Experimentally we observed that
if s is smaller than 1.74 ± 0.02, this perturbation does
not destroy the elliptic point in the phase space.
Nevertheless, a great number of lock-ins between the oscillations of the compass at 60o and the perturbation
at 2 w can occur. They appear when 60o = (m.2 cv/n).
In this case the system has to be acted on in order that they may set in. When s is increased these lock-ins appear with a well-defined hierarchy corresponding
to increasing 60o; for instance, when s >, 0.88, or
s ~ 1.05 or s > 1.36 we find that 60o is respectively
2 60/5, 2 w14, 2 w/3 t. In the case of the lock-in with (2 60/2) which appears when s >, 1.74, the elliptic point splits and the compass, which until then was
oscillating at the frequency (2 wl2 n) from side to side
of the rotating field, now oscillates with the frequency (60/2 n). The experimental evidence for this phenome-
non is slightly complicated by the rotational motion of the compass : this rotation induces a signal whose frequency itself is (w/2 n). We have, in fact, to monitor
the magnitude of the second harmonic whose sudden
growth reveals the beating of the rotation at 60 and the oscillation also at w. From this bifurcation, the same scenario repeats itself on a smaller scale and possible lock-ins will now affect the oscillation motion at 60. The two elliptic points are stable until s = 1.93,
when a new bifurcation with period doubling occurs.
It simply corresponds to the occurrence of the subhar- monic (2 w/4) (Fig. 2a) on the same scenario, but this
time the scale of the motion becomes very small and the corresponding parameter variations become extre-
mely restricted. Experimentally it is difficult to increase
this parameter without getting a lock-in 1/3 (2 c~/12),
which rapidly leads the motion of the compass to
large scale chaos (for s ~ 1.94). One fascinating pro-"
perty of this device is that it seems that the destabili- zation of all the resonances follow the same scenario.
e) In fact, dissipation results from solid friction, so this model is somewhat crude.
L-539 CHAOTIC BEHAVIOUR OF A COMPASS
Fig. 2. - Pick-up coil signal spectra. a) After two period doublings;
b) Chaotic phase; c) After a new lock-in.
For example with resonance B, M/2 appears for
s = 0.92, w/4 for s = 1.04.
3. Study of chaotic motion. - The occurrence of chaos from a lock-in appears in a first phase during
which the system is nearly locked but exhibits abnor- mal amplitudes. It slowly escapes, from the lock-in,
after about ten periods. This phase is followed by a phase of large scale chaos where the motion of the compass appears totally erratic. The compass rotates a few times along with one of the fields, then suddenly
reverses its rotation, stops, starts again and so on. We
think that such a motion illustrates the large scale stochasticity of Hamiltonian systems since the rota- tion inversions indicate that the chaotic motions are
of large extension in the phase space. Further, these
inversions may occur after a few periods of excitation
only, which is a very short time compared with the
time related to the dissipation (typically 100 periods).
Beside this, the nature of this chaos seems to be Hamil- tonian as shown by the spectra of figure 2b where it can be seen that the noise merges at once in all the spectra, which is contrary to the usual experimental
observations in dissipative systems [4]. Nevertheless the dissipation plays a fundamental role : this kind of chaos does not seem to be stable and the system finally finds a new lock-in which may be very complex
as figure 2c shows. This illustrates the attractive pro- perty of the elliptic points in dissipative systems.
However such chaotic phases exhibit a great variety
of durations, under the same conditions, from five
minutes to a few hours ! We also notice that new resonances appear at a finite s value. In the case of
figure 2c : s ~ 1.92.
We think that, even though the large scale stochas-
ticity is a property of Hamiltonian systems, it also
occurs in a transient way in slightly dissipative systems.
The experimental device that we describe allows quan- titative comparison with new theories in both dissi-
pative and Hamiltonian systems. Thus we hope, that
in a near future, we will be able to extract an experi-
mental value for the 6 exponent of Feigenbaum [5]
in both cases.
Acknowledgments. - We are grateful to D. Escande,
F. Doveil, S. Aubry, Y. Pomeau, P. Berge and
M. Dubois for stimulating discussions. We also thank M. Labouise and B. Ozenda for their technical assis- tance.
References
[1] ESCANDE, D. F. and DOVEIL, F., Phys. Lett. 83A (1981) 307.
[2] CHIRIKOV, B. V., Phys. Rep. 52 (1979) 263-379.
[3] ARNOLD, V. I. and AVEZ, A., Ergodic problems of classical
mechanics (Benjamin, New York) 1968.
[4] DUBOIS, M. and BERGÉ, P., J. Physique 42 (1981) 167.
GOLLUB, J. P. and BENSON, S. V., J. Fluid. Mech. 100 (1980) 449.
LIBCHABER, A. and MAURER, J., NATO-ASI Proceedings, 1981,
to be published.
[5] FEIGENBAUM, M. J., Phys. Lett. 76 (1979) 375.
[6] MCLAUGHLIN, J., J. Stat. Phys. 24 (1981) 377.
[7] HUBERMAN, B. A., CRUTCHFIELD, J. P. and PACKARD, N. H., Appl. Phys. Lett. 37 (8) (1980) 750.
